NOAA Technical Report ERL 407-PMEL 32

A Regional Surface Wind Modelfor Mountainous Coastal Areas

James E. OverlandMatthew H. HitchmanYoung June Han

Pacific Marine Environmental LaboratorySeattle, Washington

June 1979

u. S. DEPARTMENT OF COMMERCEJuanita M. Kreps, Secretary

National Oceanic and Atmospheric AdministrationRichard A. Frank, Administrator

Environmental Research LaboratoriesBoulder, Colorado

Wilmot N. Hess, Director

•

NOTICE

Mention of a commercial company or product does not constitute anendorsement by NOAA Environmental Research Laboratories. Usefor publicity or advertising purposes of information from this publica­tion concerning proprietary products or the tests of such products isnot authorized.

(Order by SD Stock No. 003-017-00461-9)

For sale by tbe Superintendent of Documents, U.S. Government Printing Office, WasbIDgton. D.C. 2lK02

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CONTENTS

Abstract ; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 11. INTRODUCTION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 12. THE MODEL. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 23. DETAILS OF THE MODEL. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3

3.1 Finite Difference Form " 33.2 Boundary Conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 33.3 Flooding 43.4 Entrainment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 43.5 Initialization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4

4. SIMPLE EXPERIMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 44.1 Onshore Flow with Flat Topography. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 54.2 Offshore Flow with Flat Topography. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 64.3 Modification of Flow by a Coastal Mountain : 7

5. SIMULATION FOR PUGET SOUND/STRAIT OF JUAN DE FUCA 95.1 Regional Description. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 95.2 Data Sources " 95.3 Model Simulations 11

5.3.1 Cyclonic Storm System 135.3.2 Interior High Pressure 135.3.3 Offshore High Pressure 26

6. ACKNOWLEDGMENTS 327. REFERENCES " 32APPENDIX: Derivation of Boundary Layer Equations 33

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A REGIONAL SURFACE WIND MODELFOR MOUNTAINOUS COASTAL AREAS*

James E. Overland, Matthew H. Hitchman,and Young June Han

ABSTRACT. A mesoscale numerical model of the planetary boundary layer (PBL) wasmodified for application to mountainous regions along the northwestern coast of the con­tiguous United States and the southern coast of Alaska. The model treats the PBL as a one­layer primitive equation system solving for boundary layer height, potential temperature,and the two components of horizontal velocity. Input parameters are the large-scale geo­strophic wind pattern and the stability of the air mass. Experiments with a cross-section ver­sion of the model were performed to assess its response to variable terrain, differential heat­ing, and differential roughness at the coast for a domain containing both a flat coastal plainand low coastal mountains. The complete model was applied to three quite dissimilar mete­orological situations for the Puget Sound/Strait of Juan de Fuca system in northwesternWashington state. The model is specifically useful in suggesting the relative roles of inertiaand topography.

1. INTRODUCTION

A major limitation of coastal marine mete­orology is the inadequate specification of the localwind field at the spatial resolution necessary to re­solve wind drift, local waves, and vessel or oil­spill leeway. Typically, this is due to the difficultyin estimating nearshore wind fields directly fromlarge-scale synoptic patterns or from widelyscattered and often unrepresentative wind meas­urements. Near the coastline, topography and dis­continuities in surface roughness and heating giverise to significant mesoscale variations. Strongageostrophic winds exist in the passes of the south­eastern Alaskan coast and are attributed to chan­neling around islands. The open coast is also sub­ject to anomalous winds caused by high coastalmountains. Of particular importance are windsblowing off the land, called katabatic winds,forced by the contrast of warm ocean tempera­tures and cold temperatures 50-100 km inland.Farther south, in the Puget Sound basin in Wash­ington, forecasters are aware of a quiet zone of re­duced winds in the lee of the Olympic Mountains.This zone changes location as a function of the off­shore wind direction. Sea breeze circulation is an

additional example of coastal wind modification.This report documents the combined use of a

numerical meteorological model and a field meas­urement program to explore regional wind pat­terns. Within the context of its formulation, themodel can be used to assess how changes in large­scale flow, surface parameters, and assumed dy­namics affect the wind pattern in a limited region.A major goal is the ability to infer local winds andsmall-scale spatial variations in wind fields fromthe large-scale flow pattern for locations wherelong-term observations are not practical.

We' have chosen to adapt the mesoscalenumerical model of the planetary boundary layer(PBL) proposed by Lavoie (1972, 1974; see alsoKeyser and Anthes, 1977). Lavoie treats the PBL,typically 0.5 to 2.0 km deep, as a one-layer, verti­cally integrated primitive equation model. Themodel solves for boundary layer height, potentialtemperature, and the two components of hori­zontal velocity, throughout a limited region.Large-scale geostrophic wind, surface elevation,temperature, and the stability of the air in thelayer above the PBL are specified as boundaryconditions. Air temperature and PBL height arespecified along the inflow boundaries. The localresponse is calculated by specifying smooth initial

·Contribution No. 366 from the NOAA/ERL Pacific Marine Environmental Laboratory.

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2. THE MODEL

Figure I.-Model-defined parameters of height, velocity, andpotential temperature.

The atmosphere is represented by four layersdefined by changes in the lapse rate of potentialtemperature, as shown in fig. 1. The layer in con-

I!..

(3)

(1)

a(h~~)(} + v(h-D)v(}

= E(}+ - CH Iv I ((}-(}s).

= -(h-D)Fj+ g(h-D)A() V h

(}o

+ g(h-D)2 V() + EV+-CDlv[v, (2)2(}o

a(h-D)v + V(h-D)vv + (h-D)fkxvat

The right side of the mass conservation equa­tion (1) represents the recruitment of mass into thePBL through entrainment of the overlying fluid atrate E. The height of the top of the surface layerabove sea level is indicated by D, so that h-D is thelocal PBL thickness. The thickness of the surfacelayer is o. In the momentum equation (2), the sec­ond term is inertia; f is the Coriolis parameter; g isgravity; (}o is a reference temperature; v+ is the ve­locity at the base of the inversion layer (entrainedinto the PBL at rate E), and CD is the surface dragparameter. The temperature increase between thePBL and the inversion layer is A(}. The air stabilityassociated with the inversion is thus modeled as ajump condition in density. Fi represents the uni-

tact with the surface is a constant stress or surfacelayer assumed to be represented by a logarithmicvelocity profile. The upper limit of this layer istaken to be 50 m. Above the surface layer is theplanetary boundary layer (PBL), represented byvertically integrated values of velocity and poten­tial temperature. The PBL is capped by a densitydiscontinuity, which parameterizes the restoringforce of an inversion layer of stable air above thePBL. The PBL is the only layer that is explicitlymodeled. The model specifies four dependent vari­ables: the PBL height, h, identified with the inver­sion base in unstable or neutral stratification; thePBL potential temperature, (); and the two compo-

nents of the vertically integrated wind velocitywithin the PBL, u and v, represented here by thevector v. The governing equations for conserva­tion of mass, momentum, and heat result fromvertically integrating the primitive equations forthe PBL, treating the lower atmosphere as a Bous­sinesq system. Interactions with the surface layerand upper atmosphere are parameterized. The re­sulting equations (see appendix) reduce to

ahat + V (h-D) v = E,

SurfaceLayer

InversionLayer

TransitionLayer

PlanetaryBoundary

Layer

FreeAtmosphere

50 m

----r---- ---8 300 - 2000 m

h

values of wind, temperature, and PBL height, andthen time-stepping the equations of continuity,momentum, and heat conservation until a state ofequilibrium is obtained. The system allows esti­mation of mesoscale wind variatiol)s caused bycontrasts in heating and roughness of land andwater, modification of the down-wind environ­ment by advection, and channeling by topog­raphy for a given static large-scale pressure pat­tern. Since the model considers only one layer,processes that depend upon vertical structure can­not be directly resolved. For example, questionsremain on the adequacy of the model to representsea breeze circulation without explicitly resolvingreturn flow aloft. However, the model is wellsuited to estimating wind patterns in mountainousregions with strong orographic control.

The Puget Sound/Strait of Juan de Fuca re­gion in northwestern Washington was chosen fora test basin because there was a fairly comprehen­sive data set available for comparison. Since themodel is quickly dominated by complex topog­raphy, several cases with simple geometry, basedon a topography similar to that of western Ore­gon, are included in Section 4 to build confidencein interpreting more complicated results. Thequestion of the type and quality of large-scalepressure field input is also addressed by compari­son of prepared sea level pressure analysis withcomputer-generated objective analysis from theNational Meteorological Center.

DD-8-----L--~-----/­

8s

2

form pressure gradient associated with the back­ground large-scale flow (the major input to themodel), while the next two terms consider pres­sure gradients induced by the local variations inPBL height and temperature. In the absence ofmesoscale variation, (2) reduces to a geostrophicbalance modified by surface drag. The right-handside of the heat equation (3) indicates that the PBLcan be warmed by entrainment at the top of thePBL (0+ being the temperature ·at the base of theinversion) or by surface heating proportional tothe difference between the PBL air temperature, 0,and the surface temperature, 05.

The wind velocity v is an average for the en­tire PBL. Since almost all wind shear is confined tothe surface layer, the model wind can be taken asnearly equal to the wind at 50 m elevation. At thislevel the wind speed is approximately 15 % greaterthan the wind measured at the normal anemom­eter height of 10 m. Corrected for height in thismanner, the model winds should correspond to3D-min averaged anemometer winds, which arenot unduly influenced by surface features smallerthan the mesh length of the model for a well-mixedPBL.

--

) )

- --

( ()

- -Figure 2.-Staggered mesh for primary variables. Mesh loca­

tions specify the following variables: x-temperature (11) andPBl height (h); o-v component of velocity; .-u compo­nent of velocity.

For domain sizes on the order of several hun­dred kilometers it is important to emphasize thegravitationally controlled circulation, which re­quires specification of either boundary layerheight or inflow velocity. Along inflow bound­aries over the ocean, we have chosen to specifyconstant PBL height, hi, and air temperature, Oi.These values are held fixed for all time. Inflowboundaries over land specify the PBL height andtemperature as

subject to a minimum PBL height. This minimumwas set at 300 m. After the h values are set by (4),they are smoothed twice to remove the influenceof rapid variations in the ground elevation D. Themodel needs to be rerun on a case-by-case basis,adjusting the constant a to minimize the influenceof the open boundary on the height field at inte­rior points. The authors are currently experiment­ing with setting the PBL height along inflowboundaries from the results of a one-dimensionalmodel. At outflow boundaries we follow Lavoieby setting the PBL height and potential tempera­ture at their upstream values.

Specifying the momentum flux through theopen boundaries for the non-linear advectionterms in the momentum equations must be donewith care, because advection in a limited domain

3. DETAILS OF THE MODEL

3.1 Finite Difference Form

The chosen grid is a single Richardson lattice(fig. 2) in which the u and v components arestaggered relative to the height field and eachother. This approach is optimal for gravity waves.This lattice also eliminates over-specification ofboundary conditions, a difficulty with Lavoie'soriginal formulation. The flux form of the advec­tive terms maintains conservation of scaler quanti­ties. Upstream values, rather than centrally aver­aged values, for advected quantities are chosen tomaintain the transportive property, which guar­antees one-way flow of information.

3.2 Boundary Conditions

Specification of boundary conditions forlimited-area integrations of the primitive equa­tions is a formidable task. One advantage of thepresent approach is that constant values on theboundaries can be specified, with the integrationrun until all the ringing of the time-dependentmodes is frictionally damped.

3

h = hi + aD, a = 0.5,o= OJ,

(4)

is significant. Several options for inflow velocitieswere investigated, including specifying thelaterally homogeneous solution for the given geo­strophic wind and drag coefficient. This provedunsatisfactory because the imbalance between theboundary values and the internal values influ­enced by orography caused severe geostrophic ad­justment problems throughout the model domainand resulted in large deviations in the height field.Our final choice is to assume zero gradient condi­tions on the velocity components at the inflowboundary. This assumption results in determina­tion of the values at the first interior point by thelocal dynamic balance. This decision is consistentwith the limited data input available and the desireto resolve orographic control interior to themodel. Since upstream differencing is used for mo­mentum advection, only minor difficulties are en­countered at outflow boundaries.

3.3 Flooding

In the presence of high mountains or lowmean velocities, the top of the marine inversionlayer may actually intersect the topography. Forthe vertically integrated model, this is equivalentto forming an island.

In the cases studied by Lavoie it was not nec­essary to resolve this feature, but it must be re­solved for such cases as those of Puget Sound andthe high Alaskan coastal mountains. In the presentmodel flooding is accomplished by selectively re­moving a grid point if the PBL depth falls below apreset value, and adding points if the surroundingPBL heights are great enough to increase the PBLdepth above a minimum. Since adding or drop­ping points creates new internal boundary condi­tions, flooding increases the relaxation time tosteady state by a factor of three.

3.4 Entrainment

In our initial application to mountainous re­gions we will assume that an oceanic PBL heightcan be specified a priori and, for the time intervalnecessary for a parcel to flow through the domainof the model, that no significant modification iscontributed directly through entrainment, Le., E isset to zero. Entrainment can be added to this typeof model (Stull, 1976, for example), but representsa major complication and is of secondary impor­tance relative to the influence of large topographicfeatures.

3.5 Initialization

The values of parameters and input condi­tions in table 1 are used in subsequent model runs.

The background large-scale pressuregradient, Fi, is calculated to balance the specifiedgeostrophic wind, vg , The PBL- height is initializedby hi and velocities are initialized by 70% of thegeostrophic wind.

4. SIMPLE EXPERIMENTS

In the sections to follow, complex topog­raphy dominates the flow field through the over­lapping influence of several mountains and thecontrasts between land and water. These all con­tribute to local modification of the wind field, Toaid in interpretation of more complex results, wefirst describe several experiments with simpletopography, isolating particular physical pro­cesses. The examples use a one-dimensional ver­sion of the model (Le" north-south derivatives areset to zero) with the parameters given in table I,The experiments consider either a flat topographywith a land/ocean discontinuity or a mountainbarrier 700 m in elevation located adjacent to the

Table I.-Input parameters

"

Even in the absence of mountains, determina­tion of the PBL height is a complex problem. Forunstable boundary layers the height cannot be ex­plicitly determined, but is governed by a rateequation that considers free and forced convec­tion, large-scale subsidence, shear instabilities,and solar radiation. The importance of entrain­ment is problem-dependent and we can supposethat it is more significant in the Gulf of Alaska inwinter, for example, with cold air outbreaks overwarm water, than in Puget Sound.

4

Parameter

g

fCo (water)CH (water)CD (land)CH (land)t..000

h,ill

Value

980.6 em see-2

1.08 x 10-4 see-'1.5 X 10-3

1.5 X 10-3

9.0 X 10-3

7.0 X 10-3

3.0K281 K600m3km13 m s-'o

coastline. The model is assumed to run from westto east, with the ocean to the west. While the lattertopography is fictional, it is roughly comparableto a slice through the Coast Range in Oregon. Thetotal domain is large (300 km) to reduce the influ­ence of the inflow or outflow boundaries. The gridmesh is 3 km. While most of the conclusions inthis section can be derived from analytic solutionsor scale analysis, we take the numerical approachconsistent with development of the two-dimen­sional model.

4.1 Onshore Flow withFlat Topography

line. Clearly, in the absence of heating and moun­tains, inertia dominates onshore flow, resulting inalmost no modification of the marine wind until itreaches the coastline.

The third example (fig. 4) is a sea breeze witha background geostrophic wind of 3.0 m S-I from290 °. The land temperature is 291 K, 10 Kwarmer than the ocean. The temperature equi­librates to 90 % of the temperature contrast100 km inland from the coast. There is little varia­tion in direction except for a delayed frictionalturning inland. The wind speed is maximum at thecoastline in response to pressure gradient inducedby the land-water temperature difference. Conti­nuity in this model requires a lowering of the PBL

a

b

Figure 3.-(a) Onshore flow with flat coastline; (b) same casewith acceleration terms set to zero.

PBL Height

PBL Height

Inertia/PressureGradient Force

r-----Wind Direction j

i... Wind Speed ~

±"'---J

Water

6001-----­

30 60 90 120 150 180 210 240 270

Distance (km)

800E

~ 290~~ 250OlQ)

"0 210

6001---.,---.,...-''+­

30 60 90 120 150 180 210 240 270

Distance (km)

E 800Water

2.0

0.0'-- _

o"§ 1.0

~ W_in_d_D_ir_e_ct_io_n_~

'7~ 13.0G=---- 1 Wind Speed J90L ,....---------

en

~ 280F

g' 240["0

'7en

13.0~

E 9.oL1000

Figure 3a shows the simplest case of onshoreflow for a flat coastline. Geostrophic wind ap­proaches from the left (270°) at 13.0 m S-I with aboundary layer height of 600 m and no contrastbetween the temperatures over land and water.Seaward, the horizontally homogeneous solutionmatches the analytical solution for a momentumintegral (Brown, 1974) with the boundary layerwind 0.96 of geostrophic and an inflow angle of17°. Coastal influence begins near the shorelineand, inland, results in a PBL height increase of260 m and a reduction in wind speed to 9.0 m S-I.

One measure of the relaxation distance for theflow to return to a near geostrophic-frictional bal­ance is given by the ratio of the magnitude of theinertia terms (uou/OX, etc.) to the large-scale pres­sure gradient force (fVg). This ratio is given as thetop curve in fig. 3a; it is largest just landward ofthe coast and is 0.1 at a distance of 100 km inland.Near the outflow boundary the solution again fitsBrown's solution for the increased drag coefficientover land. For mass continuity in a one-dimen­sional model with no entrainment, the product ofthe u component of velocity and the PBL depthmust be constant throughout the model domain.In the example of fig. 3a, conservation is satisfiedto better than 0.2 %.

The importance of momentum advection isfurther I.llustrated by contrasting 3a with 3b. Infig. 3b, the same conditions are specified as in 3a,except that the momentum advection terms are setto zero, leaving large-scale and locally inducedpressure gradients and friction as the only forces.The seaward extent of coastal influence is muchgreater. The main feature, induced by the rise inthe PBL height, is a coastal jet of 14.5 m S-I from226 0

, nearly a 65 ° change from the offshore direc­tion. Another important feature is a nearly com­plete frictional equilibrium landward of the coast-

5

O.OL.- ~

Figure 4.-Sea breeze circulation with a flat coastline.

Wind Direction

Inertia/PressureGradient Force

rn 305Ql

~ 285ClQl

'0 265

~ ~~gbb-T~e~m~pe~r~a~tu~r.:.e--dI~--------~J

.2 0.4"§

0.01...-__--

nitude of the geostrophic term at the limit of themodel, 180 km seaward of the coast. Velocitiesover land are in frictional equilibrium but theygradually increase offshore to a super-geostrophicmagnitude of 14.6 m S-I at a distance of 110 kmfrom the coast. A gradual decline is indicated nearthe limit of the model domain. For an overwaterdrag coefficient of 1.5 X 10-3, the boundary layerhas only begun to equilibrate with surface frictionwithin the model domain. One can project thatcoastal influences of offshore flow extend seawardat least 300 to 500 km. This length scale is furthersubstantiated by the land breeze case shown infig. 6, in which the ocean is 10 K warmer than theland. The air temperature increases only 3 Kovera distance of 180 km. The contribution of the landbreeze increase over the background flow is oforder 1 m S-I, compared with the sea breeze­induced increase of 3 m S-I. The length scale forthermal equilibrium of a coastal temperature dis­continuity is well beyond the domain of the modeleven for modest advective velocities on the orderof4 m S-I.

Inertia/Pressu reGradient Force.2 2.0

"§

4.2 Offshore Flow withFlat Topography

E :::~PBLHeightWater Land

300o 30 60 90 120 150 180 210 240 270

Distance (km)

height in the vicinity of the coast as.a result of theincreased velocity; the resulting slope of the PBLheight influences the winds 40 km seaward of thecoast. An interesting feature is the double peak inthe magnitude of the inertia terms.

Figure 5 shows an offshore wind for the sameparameters as in 3a. There is acceleration acrossthe coastline with a maximum 6 km offshore. Ac­celeration terms still account for 20% of the mag-

~290t Temperature V-- j

rn 280Ql

~ 310EWind Direction f ----3~ 270

'rn 7.0E windspee~ 3E 3.0

-

Wind Speed4.0,

rn

Figure 6.-Land breeze circulation with flat coastline.

1200 PBL Height

800r-_~

E400

0L....l.---I........!-.L......I--'-....l-...lL-L--L---I........!-.L......I--'-....l-...l...-L--L...Jo 30 60 90 120 150 180 210 240 270 300

Distance (km)

E 3.01--__-Inertia/Pressure

~ ~:~E Gradient Forc:f-. ...,3

3~ 260 Wind Direction

Cl~ 240

'rn 13.0~ Wind Speed t~E 9.0 t-----_%'

900~ b JPBL Height

E:~~ 1IIIIW~t~~11111o 30 60 90 120 150 180 210 240 270 300

Distance (km)

Figure 5.-0ffshore wind with flat coastline.

6

b

3.0 Inertia/PressureGradient Force

8.0

800

400

oL......J..--'-...l...-.L......J.---I..-'-...L..-L......J..--'-...L-.L......J.---I..-'-...L..-L.......l.-lo 30 60 90 120 150 180 210 240 270 300

Distance (km)

2000 PBL Height

1600

1200E

o 2.0

~1.0O.OL....- _

16.0 Wind Speed

';'C/l 12.0

E

I ::~E Wind Direction~----~

~

4.3 Modification of FlowBy a Coastal Mountain

Figures 7a, 7b, and 7c show the case of acoastal mountain, with onshore flow for threeoptions of offshore PBL height and no tempera­ture contrast. Even for moderate terrain the resultsare qualitatively very dissimilar to the flat coastalplain. All three cases show similar patterns of acoastal influence zone that extends from SO to100 km offshore. The offshore transition is notgradual, but is marked by a sharp front at the sea­ward limit as seen in the PBL height and magni­tude of the advective terms. Within this "offshorecoastal zone" the winds are reduced by as much as40% with a minimum approximately 20 to 40 kmoffshore. The winds veer to the southwest as theyapproach the coastline and accelerate on the lee

ac

400

oL.......l.--'-....L-.L......J.---'-...J.......L..-l.-J.--'-...L-.L......J.---'-...J.......L..-l..-I.-lo 30 60 90 120 150 180 210 240 270 300

Distance (km)

3.0 Inertia/PressureGradient Force

.Q2.0

~ 1.0

0.0

C/l260

~Q)

eg> 240

Wind Direction"0

16.0 Wind Speed

~';'C/l

E 12.0

8.0

2400PBL Height

2000

1600

E1200

800

PBL Height1200

E 800

3.0 Inertia/PressureGradient Force

400

0L......J.....J.......l...-.L......J.---I..-'-...L..-L.......l.--'-...L-.L......J.---I..-'-...L..-l.-J.-lo 30 60 90 120 150 180 210 240 270 300

Distance (km)

• 200~Q)

e 250

~ 210 Wind Direction

o 2.0<=~

1.0

0.0'------_

6.0 Wind Speed

1600

14.01=- ......

';"C/l 10.0E

Figure 7.-0nshore flow with coastal mountain. Offshore PDL height equals (a) 500 m, (b) 900 m, and (c) 1500 m.

7

Figure B.-Sea breeze circulation with coastal mountain.

"

PBL Height

-~---3

~

---J400

0L.-l..--'-...L.-J..-.J.--'--'--'--'L.-l......L....L.-J..-.J.--'--'--'--'---'-...Jo 30 60 90 120 150 180 210 240 270 300

Distance (km)

1200 t=-_---800

E

8.0 Inertia/Pressure0 Gradient Force~ 4.0

0.0

j ~286ETemperature f-278

~~ 320tWind Direction

~g' 240

"

~ ". 10.0~ Wi,d Spood hE 6.0

2.0

8.0 Inertia/Pressu reo Gradient Force~ 4.0

'"~ 290 Wind Direction

g' 250

"~ kWind Speed lA'", 6.0

E 2.0 --------

1200 PBL Height

800

~ 290E Temperature I ____280 -:----------V

O.OL---------~

E400

OL.....J.----'---'---'-...I...-.l-J'--'-----'---'---'-.L-l.-J---,"----'--'--'--'--'o 30 60 90 120 150 180 210 240 270 300

Distance (km)

Figure lO.-Land breeze with coastal mountain.

Figure 9.-0ffshore flow with coastal mountain.

through valleys would be enhanced by the highpressure developed on the windward side of theridge. Figures 9 and 10 show offshore flow andland breeze for a low coastal mountain. Unlike theonshore flow case with constant friction on the leeside of the mountaiI1, ,a pronounced minimum inthe PBL height does not occur when there is a re­duction in friction on the leeward side of themountain. This case strongly contrasts with theoffshore flow case for flat topography in that thereis virtually no variation in velocity seaward of thecoastline. In the land breeze case, the temperaturecontrast reinforces the down-slope flow, resultingin a maximum wind speed of 9 m S-I at the coast,reducing to 4 m S-I at 20 km offshore.

Several important qualitative results can beinferred from the one-dimensional model runs.First, the length scale for frictional and thermody­namic equilibrium over water is several hundredsof kilometers; this is consistent with wintertimeobservations of outbreaks of cold continental airover the Atlantic Ocean along the northeasterncoast of the United States. Second, in the vicinityof discontinuities, advective effects dominate.Third, the presence of even modest orographymodifies the offshore flow pattern. One can antici­pate that alongshore variations in topography arealso important. Finally, except for certain specialcases, observations made right at the coast shouldbe, at best, only qualitatively similar to the off­shore flow field.

PBL Height1600

o Inertia/Pressure~ 3.0E Gradient Force /i j

0.0 ------

~ 280 EWind Direction -J: Jg' 200 ------.,...... 1 ]"

~16.0 Wind Speed

.. 12.0

'"E 8.0

400I-------..J

0L.......1......L......l-.L.....l---L--l.-...l.-L....L-L-...l-.L.....l---L....I.......l.-L.......l......Jo 30 60 90 120 150 180 210 240 270 300

Distance (km)

800

1200E '-----

side of the mountain. They then recover to a near­frictional balance within 40 km of the lee side PBLminimum. Figure 8 shows the influence of thepresence of the mountain on sea breeze circulation(10 K temperature contrast between land and sea).In this formulation the mountain acts as an effec­tive barrier to development and emphasizes theimportance of low-level valleys in the mountainrange for the development of sea breeze circula­tion. In addition to temperature contrasts, flows

8

5. SIMULATION FOR PUGETSOUND/STRAIT OF JUANDE FUCA

A matter of primary importance is determin­ing the transport mechanism of any petroleumspilled into the waters of Puget Sound or south­eastern Alaska. Since winds have a sizable effecton surface drift, direct measurements of windsover the water are being made as part of coastalassessment programs. A goal of the regional mete­orological model is to extend the usefulness ofthese observational data sets and to enhance theunderstanding of the mesoscale atmosphericresponse.

We have selected for modeling three gener­alized examples of meteorological flow conditionsfor the Puget Sound system. Midwinter is charac­terized by a series of cyclonic storms (Section5.3.1) with strong winds from the southwestcarrying warm moist air inland over westernWashington. Another frequent winter case is thelull between storms, with high pressure to the eastof the region (Section 5.3.2) causing strongeasterly winds along the axis of the Strait of Juande Fuca with relatively light winds elsewhere. Inthe summer months, anticyclonic flow around awell-developed semi-permanent high pressure cellto the west of the region (Section 5.3.3) causes pre­vailing northwest winds offshore along the west­ern coasts of Washington and Vancouver Island.

5.1 Regional Description

The area investigated includes western Wash­ington, the southern end of Vancouver Island, andsouthwestern British Columbia. Major featuresare the offshore ocean, Puget Sound, and theStraits of Juan de Fuca and Georgia (fig. 11). Thisregion spans the coordinates 121 oW to 126 oW and46 6 N to 56°N. Topographic data for the modelwere obtained from a master tape at the NationalCenter for Atmospheric Research (NCAR). Themesh is a grid of 5 minutes latitude and longitude,with an average elevation computed for eachsquare. The NCAR elevation data were smoothedin both directions (Shuman, 1957). Figure 12 pre­sents a view of the smoothed topographic grid asseen from the southwest.

The Cascade Mountains form a north-southbarrier to the east ranging from a low elevation of916 m at Snoqualmie Pass to a high of 4392 m at

9

Mt. Rainier, with an average height of 1800 m.The Olympic Mountains in the center of the re­gion rise gradually from the south and west to2428 m at the summit of Mt. Olympus, with anaverage height of 1600 m, descending rapidly tothe north and east. A significant area of higher ele­vation to the south is the Willapa Hills, 300 to600 m high, between the Columbia River and theChehalis River Valley. Vancouver Island is pri­marily mountainous, with heights averaging900 m, reaching 1200 min several locations.

This topography establishes one main low­level north-south passageway, extending from theColumbia River Valley through Puget Sound, andtwo low-level east-west passageways, the Strait ofJuan de Fuca and the Grays Harbor Inlet/ChehalisRiver Valley area.

5.2 Data Sources

We obtained a set of data that adequatelyrepresent the regional wind field during the periodNovember 1976 through January 1977. This set in­cludes data from routine meteorological stationreports and from an array of recording anemom­eters at strategic locations. Figure 13 and table 2provide station locations, sources, and NationalWeather Service (NWS) station symbols. Teletypedata for NWS offices and Coast Guard stationswere obtained from the Ocean Services Unit of theSeattle Weather Service Forecast Office. TheWeather Service offices and ships from the North­east Pacific typically report every 6 h. The CoastGuard stations usually report every 3 h, but mostdo not report during the night. Three MRI Model7092 Anemometers set out by the authors yieldedstrip charts, which were converted to 1-h averagesand plotted every 6 h. Data from three vector­averaging anemometers in the Strait of Juan deFuca were provided by the MESA Puget Soundproject (Holbrook and Halpern, 1978). It shouldbe noted that stations 10-17 in table 1 2. ~ well in­land; thus, local microtopography affects the airmovement there more than at the shore stations,and makes them less indicative of the generalflow. Station wind reports were mapped every 6 hfrom 0000 Greenwich Mean Time (GMT) on 27November 1976 to 1800 GMT on 26 January 1977.From these regional maps, examples of typicalweather events were selected.

For each case selected, large-scale synopticpressure maps centered on western Washingtonwere prepared from North Pacific synoptic charts.In addition, objective sea level pressure analyses

125° 124° 123° 122°

United States

"",-/"Mt. Baker

50°

48°

47°N

49°

oll>CIl(")ll>0.<t>

s::oC:J

~.:JCIl

Mt. Rainier

,1....­-/\-

122°W

":" Seattle

Cowlitz River Valley

Chehalis River Valley

125°

Kilometers

25 50 75 100I I I I

oI

47°

50°

48°

Figure 11. - The Puget Sound/Strait of Juan de Fuca region.

L

10

Figure 12.-Topographic grid used in the computations asviewed from the southwest.

Two basic regimes describe the generalweather characteristics of Decembers in westernWashington. As is typical of the latitude, a succes­sion of frontal passages from the west, varying innumber and intensity, dominates, producingstrong winds from the southwest. Between storms,high pressure builds up near the area, often in thecontinental interior, bringing clear skies and rela­tively low winds lasting for several days to a weekor more. The fall and winter of 1976 were unusualin that a persistently recurring ridge of high pres­sure (500 mbar) over the Northeast Pacific, fre­quently extending almost to the pole, allowedonly an occasional weakened frontal passagethrough the area. Surface high pressure associatedwith the SOO-mbar pattern, but displaced eastwardover the continent, dominated the Puget SoundBasin.

We selected examples from the data set de­scribed above to model both of the basic winterregimes, the typical midwinter case (cyclonicstorm system) and the frequent lull between win­ter storms (interior high pressure). In addition, wechose to use data from the same winter set tomodel the typical summer case (offshore highpressure), since, in all respects except temperature,the example selected is representative of the sum­mer pattern.

Table 2.-Stations used on local wind maps

Number Station Location Source Symbol

1 Transient Ships W2 Columbia River

Weather Ship W NNCR3 Astoria W AST-7914 Willapa Bay C 89S5 Cape Shoalwater A6 Westport C 84S7 La Push C 87S8 Quillayute W UIL-7979 Cape Flattery C 93S

10 North Point W 10511 Snider W 10912 South Olympic W 13813 Dayton W 25314 Wolf Point W 57215 Round Mountain W 74316 Lester W 45017 Little Mountain W 43218 Buoy 3 H19 Buoy 4 H20 Buoy 2 H21 Port Angeles C NOW22 New Dungeness C 96S23 Point Wilson C 53S24 Point Wilson A25 Smith Island26 Friday Harbor C S1927 Victoria W VI-20028 Patricia Bay W YJ-79929 Cassidy Airport W CD-89030 Comax W QQ-89331 Vancouver W VR-89232 Abbotsford W XX-10833 Point No Point C 97S34 WestPoint C 43S35 WestPoint A36 AlkiPoint C 91S37 Sea-Tac Airport W SEA-79338 Point Robinson C 99S

KEYC Coast Guard station reportA PMEL land-based anemometersH PMEL buoy-placed anemometersW National Weather Service station report

on the Limited Area Fine Mesh Model (LFM) gridwere obtained for the region from the NationalMeteorological Center. We will compare the ob­jective analyses on the 160-km mesh to the hand­drawn charts to determine whether LFM input isadequate for the regional model. Upper-air sound­ing data were available from Quillayute station onthe Washington coast; weather ship Papa, locatedat SooN, 145 oW; Sea-Tac airport, south ofSeattle; and Portage Bay in Seattle. The pressureanalysis maps show pressure in millibars, writtenout to the units place on isobars and to the tenthsplace at stations. In the latter the first two digitsare deleted; e.g., 236=1023.6 mbar. Wind is givenon these maps as barbs (one full barb = 10 kn).On the local wind maps, direction and speed to-

11

gether are given at stations; e.g., 2813from 280 0, speed 13 kn.

5.3 Model Simulations

wind

Willapa Hills

47°N

48°

50°

49°

•15

122°W

122°

14

•123°

123°

13.

Chehalis River Valley

12.

124°

Olympic Mountains

125°

50°

49°

Vancouver Island

48°

47°

Figure 13.-locations of anemometer stations that collected data for December 1976-January 1977.

l12

5.3.1 Cyclonic storm system

The front that approached the coast at 0000GMT, 8 December 1976 (fig. 14), turned into acold front of respectable energy as the high re­treated far to the south. The even isobars andsouthwesterly geostrophic flow before this frontare typical before the passage of a cold front.From the local wind vectors (fig. 15), one firstnotices that the flow is channeled by the Olympicand Cascade Mountains. Winds over Puget Soundare stronger and more southerly than offshore. Aregion of light winds is evident in the lee of theOlympic Mountains. There is also general steerageof the flow along the axis of the Strait of Georgia,more than a 90° deflection from the geostrophicwind. The temperature sounding at 1605 GMT on7 December at Sea-Tac shows a relatively moist,deep, well-mixed PBL with near-neutral stability(fig. 16). This is illustrated further by the fact thatthe 850-mbar flow is very similar to the surfaceflow on the LFM maps (see figs. 17a and 17b). Thehand-drawn and LFM surface maps agree well.Figures 18 and 19 for 0000 GMT, 15 December,show an additional example of strong winds fromthe southwest.

The corresponding storm situation of 8 De­cember 1976 is simulated by a model run for PBLheights of 1800 m and 900 m (figs. 20a and 20b).Geostrophic wind is 14.7 m S-I from 251°. Theoverall wind pattern for a PBL height of 1800 m ismuch smoother than that suggested by observa­tions. The pattern for the lower height, however,is about as detailed as the observed pattern. Aneddy has formed at the east end of the Strait ofJuan de Fuca near Port Angeles in each simulation.The PBL height deviations show a gentle rise overthe windward side of the mountains with a pro­nounced lee wave trough on the downwind side ofthe Olympics and Vancouver Island. With a lowinversion height, increased winds flow throughthe low point in the mountains of Vancouver Is­land and spill out over the inland waters. Ob­served winds in the east end of the Strait of Juande Fuca are less intense and more westerly thaneither m'odel run suggests. It may be that the posi­tion of the eddy and the magnitude of the pressuregradient that develops along the axis of the Straitof Juan de Fuca are very sensitive to the volume ofair channeled through Puget Sound, which de­pends in turn on the orientation of the offshoreflow. Inflow along the southern boundary is nothandled satisfactorily by arbitrary specification ofinversion height, especially at the land-water in­terface. However, this does not appear to undulyinfluence the flow in the central basin.

In the previous section it was noted that

13

inertia plays a dominant role in mesoscale circula­tions. The main difference between the two modelruns lies in whether the flow goes over the moun­tain or around it. Since observations resemblemore the case with a lower inversion, perhaps theeffective cross-sectional height of the mountains ishigher than the model-assumed average eleva­tions; the light stable stratification of the PBLshown in the Sea-Tac sounding may contribute toincreased channeling.

5.3.2 Interior high pressure

A good example of interior high pressure oc­curred at 0000 GMT on 1 December 1976. Forseveral days before and after this time, high pres­sure prevailed over southeastern. BritishColumbia, extending north and south' over theinterior plateau (fig. 21). In areas of flat topog­raphy, widely spaced isobars would suggest aweak flow outward from the high pressure centerwestward over the area. However, the local windshown in fig. 22 reveals a complex pattern witheasterly winds at the coast and calm or lightnortherly winds in Puget Sound. A very interest­ing feature is seen in the Strait of Juan de Fuca. Inshatp contrast to the weak and variable windselsewhere on the inland waters, there is a strongflow out the Strait, reaching 20 kn at CapeFlattery. This isolated jet was reported by Reed(1931) but is not specifically mentioned in more re­cent literature. Associated with these low-levelwind vectors are temperature soundings over thearea that reveal a strongly stratified regimethroughout the planetary boundary layer. TheSea-Tac sounding at 1610 GMT on 30 November1976 is shown in fig. 23. Lines of constant poten­tial temperature are also shown, indicating stablestratification throughout the boundary layer.

In the objective analyses from the NationalMeteorological Center, the absence of horizontalair flow seen at 850 mbar in fig. 24a for 1 Decem­ber, 0000 GMT, contrasts with the surface pattern(fig. 24b), which shows a light pressure gradienteast-west through the region in agreement with thehand-drawn map. The spacing on the surface LFMmap is 1 mbar, approximately equivalent to the10-geopotential meter spacing of the 850-mbarLFM map. The decoupling of the 850-mbar andsurface layer is consistent with the strong verticalstratification observed at Sea-Taco Stability re­stricts the flow to regions below the mountain topswhere the air is accelerated along the east-westpressure gradient out through the Strait of Juan deFuca and west through the Cowlitz Valley south ofthe Olympic Mountains. The winds are strongerin the strait than along the southern Washington

-

50°

130° 125°

65°

60°

55°

50°

45°

40°

35°N

I

I-Figure 14.-Sea level pressure chart, 8 December 1976, ooסס GMT.

14

50°

49°

48°

~2605

47° 4]0N,I,.

-/\-

Figure 15. -Local wind observations, 8 December 1976, ooסס GMT.

15

3042

E-= 1483.<:.Cl

.0;

I

153o

-15 -10 -5 o°C

5 10 15

700

750

800 _~.0E f850 ~~:;)UJ

900 8la:950

10001018

Figure 16.-Temperature sounding (•••• ), dew point (- - - -),and potential temperature (---) at 5ea-Tac for 1605GMT, 7 December 1976.

a b

iIi...

Figure 17.-0bjective analysis of (a) 850-mbar heights and (b) sea level pressure on 8 Dec.,mber 1976, 0000 GMT.

16

'l185

1020

115

331

'\

066

H

Figure lB.-Second example of strong onshore flow from southwest, 15 December 1976, 1800 GMT.

17

1817

2025

. d observations for fig. 18.. 19 -Local wmFigure .

18

r1907

~""3005

50°

1111111rr r

IIITrrrrrrrrrTrr rrInrrrTInTlrrTnTlTlrrTT1I111TTTITT11111Trr1TTT

a b

Figure 20.-Velocity vector plots of model winds for southwest flow. Offshore PBL height equals (a) 1800 m and (b) 900 m.

'The influence of temperature in the form of land-water tem­perature differences was neglected in all Puget Sound cases.Thus, it was not necessary to carry the heat equation in thecalculations.

was initialized by a geostrophic wind of 4.8 m S-I

from 144 0 and a low PBL height of 0.6 km as rep­resentative of stable conditions throughout thelower troposphere. The major features are lightwinds in the central basin, weak easterly flowalong the coast, and accelerating easterly flowdown-gradient through the Strait of Juan de Fuca,

Table 3.-Model input parameters'

E 1 Dec OOZSW, 8 DecooZSW, 8 DecooZNW 9Dec 12Z -

T.(K)

0.6 9.8 273.01.8 7.0 289.50.9 7.0 289.51.0 5.2 280.1

144 0

260 0

260 0

321 0

Wind hi IiliDirection (km) (K)

4.814.714.716.2

CeostrophicSpeed(m/s)

Date(1977)

WindType

coast because the down-gradient acceleration isuninhibited by surface friction. Another curiousfeature is that the winds in Puget Sound properflow south, in the opposite direction to those in­ferred from the surface geostrophic wind. A sec­ond example of winds under the high pressure re­gime is seen in figs. 2S and 26, where high pressurehas built up rather rapidly between frontal pass­ages. The local stations again reflect the widelyspaced isobars with easterly winds on the coast,calm in the sound, and acceleration along theStrait of Juan de Fuca.

Figure 27a shows the wind pattern generatedby the model corresponding to the case of 1 De­cember 1976. Although the boundary layer is notwell mixed as assumed in Section 2, we believethat we can simulate the forced channeling for theeast wind case by assuming a very shallow PBL inthe model, capped by very strong stability. Inputparameters are summarized in table 3. The model

19

Fiwure 21. -Sea level pressure analysis, 1 December 1976, 0000 GMT.

20

i

L

1905 J1909J47° 47°

N,I,...-/\-

Figure 22.-Local wind observations, 1 December 1976,0000 GMT.

21

-

3149 700

750

950

800 -;:­os.0E

850 ­~::Jen

900 :get

1000

1029

105o

.....~ 6> ....\ ~ -.." <".90 ....

'" 1- ....I i

.I :( .

_J ..J~ ....., .:

)i1/

-15 -10 -5

I 17201:OJ.0:;I

°C

Figure 23.-Temperature sounding (.••• ), dew point (- - - -),and potential temperature (---) at Sea-Tac for 1610GMT and wind at Portage Bay for 2015 GMT, 30 November1976.

La b

Figure 24.-0bjective analysis of (a) 850-mbar heights and (b) sea level pressure on 1 December 1976, ooסס GMT.

22

55°

45°

125°

125°

@233

295/

120" W

L

305@

120"

55°

50°

45°N

Figure 2S.-High pressure to the northeast of Puget Sound, 21 December 1976,1800 GMT.

23

'3125

Figure 26. -local wind observations for pressure field shown in fig. 25.

24

2709

50°

L

I

\

I.,~, , *' ~

~-~-,

a

Figure 27a.-Velocity vector plot of model run for east wind case.

similar to the conditions shown in figs. 22 and 26.As the flow in all channels is out of the PugetSound basin, this case could not be run to steadystate. In the prototype the outflowing air is re­placed by subsidence associated with the synoptichigh pressure. Subsidence is not included in themodel to balance the falling PBL height; fig. 27a isthe model-estimated wind field when the interiorPBL height has reached 400 m after 4 h and is fall­ing at a constant velocity. To increase the resolu-

25

tion in the main area of interest, the Strait of Juande Fuca, the grid length was reduced to one-half ofits previous value in the north-south direction,and the domain was also reduced to see whetherthe model could be sectionalized (fig. 27b). Goodagreement is obtained in the strait. Contrary tothe inference from observations, at the east end ofthe Strait of Juan de Fuca a more geostrophic flowoccurs if the southern end of Puget Sound is notincluded in the model domain.

... ... , , , ,, , ,

... "",

-10 ms-1

Figure 27b.-Velocity vector plot of model run for east wind case with increased north-south resolution.

5.3.3 Offshore high pressure

The front depicted in fig. 25 was the weakestof four crossing the region in December 1976. Fora day following the 8 December front and a dayfollowing the 22 December front, a cell of highpressure existed off the coast of Oregon andNorthern California and brought strong north­westerly winds through Washington as part of ananticyclonic circulation. Except for temperatureeffects, this pattern is typical of summertime con­ditions in the region. The hand-drawn pressuremap of 1800 GMT, 23 December 1976, shows arelatively uniform pressure gradient from offshoreinland to Vancouver, B.C. (fig. 28). The local ane­mometer readings (fig. 29) reveal the effect oftopography on a northwesterly geostrophic wind.Strong channeling is indicated in the Strait of Juande Fuca with variable winds in the lee of theOlympic Mountains. It is interesting that for this

26

case and for 1200 GMT, 9 December 1976 (figs. 30and 31), there is a southerly flow in lower PugetSound in the lee of the Olympics, but only at thesurface. Figure 32 shows the wind sounding for1400 GMT, 9 December, at McChord Air ForceBase, and the Quillayute temperature sounding.The LFM maps (figs. 33a and 33b) concur with thehand analysis in showing a northwesterly geo­strophic flow.

Figure 34 shows the model velocity field fornorthwest winds. Channeling is indicated in theStrait of Juan de Fuca and especially in the Straitof Georgia. Height deviations are less intense thanfor the southwest wind case, although the velocityfield indicates that the lee wave eddy is still amajor feature. A southerly tendency is indicatedin the lower Puget Sound trough where the flow isparallel to the pressure gradient below the ridgecrests.

I,L

077

./236

264

45°

102745°

H : '

:' 247

~~ Jr 208

235

/" 251

260

Figure 28.-Sea level pressure chart, 23 December 1976, 0800 GMT.

27

50°

Figure 29.-Local wind observations, 23 December 1976,1800 GMT.

28

/ 2004

50°

iI\L

130° 125° 120° 115° 1100 W

60°

55°

50°

45°

40°

35°

Figure 30.-Sea level pressure chart, 9 December 1976, 1200 GMT.

29

\3220

Figure 31. -Local wind observations, 9 December 1976, 1200 GMT.

30

2704

IIL

105

3000,

700

lilt. 750

~t.-. 800

I 1475

~, .0

850 .s:E

~Cl"0; :J

IIII ll... III

900 Q)

"-.. l1:

'--

"- 950

165 "--I 1000

0 ./ 1015

Figure 32.-Temperature sounding (•••• ), dew point (- - - -),and potential temperature (---) at Quillayute (Washing­ton coast) for 1200 GMT and wind at McChord Air ForceBase (Puget Sound) for 1400 GMT, 9 December 1976.

-a b

Figure 33.-0bjective analysis of (a) 850-mbar heights and (b) sea level pressure on 9 December 1976, 1200 GMT.

31

Figure 34.-Velocity vector plot of model winds for northwestflow.

6. ACKNOWLEDGMENTS

This study was supported jointly by theMarine Ecosystems Analysis (MESA) Puget SoundProject and the Outer Continental Shelf Environ­mental Assessment Program (OCSEAP) to assistin providing wind field information for oil spilltrajectory calculations, and by the Marine Ser­vices Program at the Pacific Marine Environ­mental Laboratory (PMEL), which aids NationalWeather Service marine prediction along the westcoast and the Gulf of Alaska.

We wish to thank Robert Anderson and hiscolleagues at the Seattle National Weather ServiceForecast Office for their collaboration in the fieldprogram and in compiling the routine observationdata sets, and Jim Holbrook of PMEL for makingavailable his anemometer observations and dis­cussing their implications.

32

7. REFERENCES

Brown, R. A. (1974): A simple momentum integralmodel. J. Geophys. Res. 79:4076-4079.

Holbrook, J. R., and D. Halpern (1978): Variability ofnear-surface currents and winds in the western Straitof Juan de Fuca. Pacific Marine EnvironmentalLaboratory, Seattle, Washington. Unpublishedmanuscript.

Keyser, D., and Anthes, R. (1977): The applicability ofa mixed-layer model of the planetary boundary layerto real-data forecasting. Mon. Wea. Rev. 105:1351­1371.

Lavoie, R. (1972): A mesoscale numerical model oflake-effect storms. J. Atrnos. Sci. 29:1025-1040.

Lavoie, R. (1974): A numerical model of tradewindweather on Oahu. Mon. Wea. Rev. 102:630-637.

Ogura, Y., and N. W. Phillips (1962): Scale analysis ofdeep and shallow convection in the atmosphere. J.Atrnos. Sci. 19:173-179.

Reed, J. (1931): Gap winds of the Strait of Juan de Fuca.Mon. Wea. Rev. 59:373-376.

Shuman, F. G. (1957): Numerical methods in weatherprediction, II: smoothing and filtering. Mon. Wea.Rev. 85:357-361.

Stull, R. B. (1976): Mixed-layer depth model basedupon turbulent energetics. J. Atrnos. Sci. 33:1268­1278.

I

l

APPENDIX: Derivation of Boundary Layer Equations

Since 11"0 is a function of z only, we can rewrite eq.(AI)

We shall write the equations of motion fordeviation from a steady reference state. If the ref­erence state changes only very little with height, itis possible to use the Boussinesq approximation,but with potential temperature as the thermalvariable (Ogura and Phillips, 1962).

The momentum equation is

au + - - au + fk- - + ()at V'VV = w az xv Cp oV1I"

av ~-+ - V- + uVat v· v W iii

+ fhv + cp()o V 11" "

a (~.)=-aw vw ,

(A6)

We shall use eqs. (A3), (A4), (AS), and (A6) fordescribing the flow field in the well-mixed layer.

We now integrate (A4) and (A6) through themixed layer. The basic equa tions then become

hf V1I""dzD

= -(v'w'h-v'w's)/(h-D),

~~ + v.V() = - (w '()h ' - w '()s ')/(h-D).

_ a(~)--- v wazwhere (P)x

11"= Po ,x = R/cp.

The hydrostatic equation is

a11"cp()az = -g.

The equation of continuity is

- + aw 0v·v az = .

(AI)

(A2)

(A3)

av -- + V·VV + fkxv +atcp()o

h-D

(A7)

(A8)

The first law of thermodynamics is approximatedby

In addition, the mass continuity equation, by defi­nition, can be written

If we define 11"0 such that

In these equations u is Reynolds' averaged hori­zontal velocity vector, v' is the deviation ve­locity, () is potential temperature, and ()o is thepotential temperature of the reference state (con­stant). The other terms are defined in the usualmeteorological sense.

We simplify the hydrostatic equation (A2) inthe following way:

cp ~: = --~ =-g~o (l-~:'}where (),,' = () - ()o.

where E is the net entrainment rate at which thewell-mixed layer gains mass from the free at­mosphere.

Using the hydrostatic equation, we evaluatethe vertically integrated pressure gradient force:

(A9)

(AlO)

~~+ V·(h-D)v = E(A4)a() + _.n() + a() = _ ~ (------;-()')at v v w az az w .

then

(All)

-

where subscript H denotes the top of the modelatmosphere.

For the convenience of finite differencing, eq.(A8) is written in a flux form:

Jt (h-D)() + V·(h-D)v()-()E

-- --= -(W'()')h + (w'()')s.

g ()"()o ()o

33

In deriving the equation, eq. (A9) was used. Themomentum equation (A7) was also put in fluxform.

Integrating eq. (AI) and (A4) across the jumpbetween the PBL and inversion layer using Leib­nitz' rule, we obtain relations

(v 'w ')h = -Eav

and (W'(J')h = -Ea(J

(All)

(A13)

The flux form of (A9) with the substitution of(AID), equations (A9) and (All), and entrainmentrelations (All) and (A13) form a closed set ofequations, (1), (2), and (3) in the text, given thatthe entrainment rate can be parameterized interms of the mixed layer variables.

34

-{:{ u. s. GOVERNMENT PRINTING OFFICE: 1980-677-09611261 REGION NO.8

iIiIL.